Q) The angles of depression of the top and the bottom of a 8 m tall building from the top of a multi-storeyed building are 30° and 45° respectively. Find the height of the multi-storeyed building and the distance between the two buildings.

Ans: 

Step 1: Let’s start with the diagram for this question:

Here we have multi-storeyed building AB of height H (we assume) and PQ as 8 m high building.

Angle of depression from A to P and Q are given.

We need to find height H and the distance between the two buildings, D.

Let’s make a simplified diagram of the same for our better understanding:

Step 2:

In Δ ABQ, tan Q = tan 45° =

∴ 1 =

∴ H = D …… (i)

Step 3:

In Δ ACP, tan P =

Since, AC = AB – CB

∴ AC = AB – PQ

∴AC = H – 8

and CP = BQ

∴ CP = D

∴ tan 30° =

∴

∴ D = √3 (H – 8) …. (ii)

Step 4:

By comparing equations (i) and (ii), we get:

H = √3 (H – 8)

∴ 8 √3 = H √3 – H = H (√3 – 1)

∴ H =

∴ H =

∴ H =

∴ H =

∴ H = 4 (3 + √3)

Therefore, the height of multi-storeyed building is 4 (3 + √3) m

Note: By substituting values of √3 = 1.732, we get:

∴ H = 4 (3 + 1.732) = 4 x 4.732 = 18.93

Therefore, the horizontal distance is 18.93 m

Step 5: 

From equation (i), we have D = H

∴ D = H = 4 (3 + √3) m

Therefore, the horizontal distance is 4 (3 + √3) m

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